Concrete Nonsense http://concretenonsense.wordpress.com A group blog about mathematics Mon, 09 Nov 2009 15:30:58 +0000 http://wordpress.com/ en hourly 1 http://www.gravatar.com/blavatar/2b77f97368a020180b21c93e224b3ae1?s=96&d=http://s.wordpress.com/i/buttonw-com.png Concrete Nonsense http://concretenonsense.wordpress.com Chow rings and K-theory http://concretenonsense.wordpress.com/2009/11/09/chow-rings-and-k-theory/ http://concretenonsense.wordpress.com/2009/11/09/chow-rings-and-k-theory/#comments Mon, 09 Nov 2009 15:27:19 +0000 Steven Sam http://concretenonsense.wordpress.com/?p=758 ]]>

I want to write a post about the set-valued tableaux of Buch and how they are related to Schur polynomials. But since these two things are related to the K-theory and Chow ring, respectively, of the Grassmannian, I thought I would write a post explaining some basic generalities between the Chow ring and K-theory. Let X be a variety over an algebraically closed field K.

First let’s define the Chow groups. We first form the k-cycles Z_k(X) to be the free Abelian group spanned by the k-dimensional subvarieties of X. Let [V] be the basis element corresponding to a subvariety V. Pick a subvariety W of X of dimension k+1, and a nonzero rational function f/g defined on W. If V is a codimension 1 subvariety of W, let \mathcal{O}_{W,V} be the ring obtained by taking the ring of polynomial functions on W and inverting all polynomial functions which are not identically zero on V. We define the order {\rm ord}_V(f/g) to be \dim_K \mathcal{O}_{W,V}/(f) - \dim_K \mathcal{O}_{W,V}/(g), where the dimension is as K-vector spaces. The divisor of f/g is given by {\rm div}(f/g) = \sum_{\dim V = k} {\rm ord}_V(f/g) [V]. We say these divisors are rationally equivalent to 0, and define the Chow group {\rm A}_k(X) to be the group of k-cycles modulo rational equivalence.

Now let’s assume that X is nonsingular of dimension n. Given subvarieties V and W of X, let Z be an irreducible component of the intersection V \cap W. Restrict to an open affine subset U of X, so that V and W are defined by ideals I and J, respectively. We define the intersection multiplicity to be the following alternating sum

\displaystyle \mu(Z; V,W) = \sum_{i \ge 0} (-1)^i {\rm length}_{\mathcal{O}_{X,Z}} {\rm Tor}^i_{\mathcal{O}_{Z,Z}} (\mathcal{O}_{X,Z} / I, \mathcal{O}_{X,Z} / J).

Set {\rm A}^k(X) = {\rm A}_{n-k}(X). We can give {\rm A}^*(X) a graded ring structure by defining [V] \cdot [W] = \sum \mu(Z; V,W) [Z], where the sum is over all irreducible components Z of V \cap W.

Now we discuss the K-theory K(X) of vector bundles. For now we don’t need to assume X nonsingular yet. We first consider the free Abelian group on isomorphism classes of vector bundles on X, modulo relations given by short exact sequences: for any sequence 0 \to E_1 \to E_2 \to E_3 \to 0, we add the relation [E_1] - [E_2] + [E_3] = 0. We endow K(X) with a ring structure via [E] + [F] = [E \oplus F] and [E] \cdot [F] = [E \otimes F] which one verifies is well-defined. Similarly, we can form the K-theory {\rm K}_\circ(X) of coherent sheaves on X, which doesn’t necessarily have a multiplication since tensoring with an arbitrary coherent sheaf need not preserve exact sequences. There is a map

\varepsilon \colon {\rm K}(X) \to {\rm K}_\circ(X)

obtained by sending the class of a vector bundle to itself, considered as a locally free sheaf. When X is nonsingular, this map is an isomorphism. The reason is that in this situation, one can always resolve any coherent sheaf by a finite resolution of vector bundles (one never needs more than \dim X + 1 such vector bundles).

In general, we can define the topological filtration of {\rm K}_\circ(X) by letting F_k {\rm K}_\circ(X) be the subgroup generated by coherent sheaves whose support has dimension at most k. Let {\rm gr}_k {\rm K}_\circ(X) = F_k {\rm K}_\circ(X) / F_{k-1} {\rm K}_\circ(X). Given a coherent sheaf \mathscr{F} and a subvariety V of X, we can define the multiplicity m_V(\mathscr{F}) as follows: on an affine open set U = Spec(A) which intersects V, V corresponds to a prime ideal P of A, \mathscr{F} corresponds to a finitely generated A-module M. So we can localize M at P to get a module over the local ring A_P, and m_V(\mathscr{F}) is the length of M_P as an A_P-module. Now for \mathscr{F} \in F_k {\rm K}_\circ(X), we can define

\displaystyle Z_k(\mathscr{F}) = \sum_{\dim V = k} m_V(\mathscr{F}) [V] \in {\rm A}_k(X).

There is a unique homomorphism \varphi \colon Z_k(X) \to {\rm gr}_k {\rm K}_\circ(X) such that [V] \mapsto [\mathcal{O}_V] and Z_k(\mathscr{F}) \mapsto [\mathscr{F}], and it factors through rational equivalence to give a map \varphi \colon {\rm A}_k(X) \to {\rm gr}_k {\rm K}_\circ(X).

In the case that X is nonsingular, we can tensor this map with the rational numbers to get an isomorphism \varphi_{\bf Q}. Then K-theory of coherent sheaves is the same as K-theory of vector bundles, and the topological filtration in this case is a filtration by subrings. In fact, after identifying {\rm A}_*(X) with the Chow ring {\rm A}^*(X), the map \varphi_{\mathbf{Q}} is an isomorphism of rings. This says that cohomology is a sort of approximation to K-theory.

Next time, I’ll specialize to the case when X is a Grassmannian, and explain the combinatorics involved with the Chow ring and K-theory. This will make the isomorphism above more clear.

-Steven

]]> http://concretenonsense.wordpress.com/2009/11/09/chow-rings-and-k-theory/feed/ 0 masnevets Exceptional sequences for the Grassmannian http://concretenonsense.wordpress.com/2009/10/26/exceptional-sequences-for-the-grassmannian/ http://concretenonsense.wordpress.com/2009/10/26/exceptional-sequences-for-the-grassmannian/#comments Mon, 26 Oct 2009 16:11:09 +0000 Steven Sam http://concretenonsense.wordpress.com/?p=733 ]]>

Let K be a field of characteristic 0, and let V be a vector space over K of dimension n, and pick k < n. Let X be the Grassmannian Grass(k, V). We’ll briefly explore the (bounded) derived category of coherent sheaves of X, denoted {\bf D}^b(X).

1. Derived category review

For those unfamiliar with derived categories, here’s a quick summary. If A is any Abelian category, set K(A) to be the category of (co)chain complexes of A with the morphisms being chain maps modulo homotopy equivalence. Chain maps which induce isomorphisms are formally inverted, and the result is the derived category {\bf D}(A) of A. Usually we only want to consider bounded complexes, or at least complexes with finitely many nonzero (co)homology groups, and in this case we denote the category {\bf D}^b(A). The category is equipped with a shift functor, which just shifts the degrees of a given complex.

One thing we can do is reformulate derived functors. Given a left exact functor F \colon A \to B, we define its right derived functor {\bf R}f \colon {\bf D}^b(A) \to {\bf D}^b(B) as follows. Given an object X in A, an injective resolution X \to I^\bullet of X becomes an isomorphism in {\bf D}^b(A) (considering X as a complex with one nonzero term), so we define {\bf R}F(X) to be the complex obtained by applying F to I^\bullet. Actually, we don’t need an injective resolution, we only need a resolution consisting of F-acyclic objects (i.e., the usual right derived functors of F vanish for them). To define {\bf R}F on a general complex C^\bullet, we need to find a double complex C^\bullet \to I^{\bullet, \bullet} which is term by term an injective resolution for each C^i (these are called Cartan–Eilenberg resolutions). Then we apply F to the total complex of I^{\bullet, \bullet}. A similar story is true for right exact functors G, so we get left derived functors {\bf L}G. For notation, the left derived functor of the tensor product is denoted \stackrel{\bf L}{\otimes}.

The replacement for exact sequences are now exact triangles, which are written as A \to B \to C \to A[1]. The relevant facts are that exact sequences of cochain complexes become exact triangles, and that if we try to calculate cohomology, exact triangles give long exact sequences. Most importantly, the total derived functors are exact in the sense that they preserve exact triangles.

We’ll need two facts. The first is the derived version of the projection formula. Let f \colon X \to Y be a proper morphism of projective schemes, E \in {\bf D}^b(X), and F \in {\bf D}^b(Y). Then we have

{\bf R}f_* (E \stackrel{\bf L}{\otimes} {\bf L}f^* F) \cong {\bf R}f_*E \stackrel{\bf L}{\otimes} F.

The second is flat base change. If we have a pullback diagram

with u flat and f proper, then we have a natural isomorphism

u^*{\bf R}f_* E \cong {\bf R}g_* v^* E

for all E \in {\bf D}^b(Y).

2. Exceptional sequences and the Fourier–Mukai transform

For this post, we’ll look for a finite set of generators \{E_1, \dots, E_N\} of {\bf D}^b(X), i.e., using only the operations of mapping cones and shifting degrees (so we’re allowed to take kernels, cokernels, and direct sums also), we can obtain the isomorphism class of every object. More specifically, we will construct an exceptional sequence. This means the following things:

  1. Every object in the sequence is exceptional: {\rm Hom}(E_i, E_i) = K and {\rm Ext}^k(E_i, E_i) = 0 for k>0.
  2. {\rm Ext}^k(E_j, E_i) = 0 for k \ge 0 and i < j.

Using the Schubert cell decomposition of the Grassmannian, it can be shown that the length of an exceptional sequence must equal the number of its Schubert varieties (and more generally, this is true for any homogeneous space for a semisimple group, see the Böhning reference below).

The main tool will be the Fourier–Mukai transform. Suppose that Y and Z are any varieties let \pi_Y, \pi_Z be the projections from Y \times Z to Y and Z, respectively. For E \in {\bf D}^b(Y \times Z), we define a map

\Phi_{Y \to Z}^E \colon {\bf D}^b(Y) \to {\bf D}^b(Z)

F \mapsto {\bf R} \pi_{Z,*}({\bf L}\pi_Y^*F \stackrel{\bf L}{\otimes} E).

Since it is a composition of exact functors, the Fourier–Mukai transform is itself exact. Also, we can let the argument stay fixed and vary the superscript to get another exact functor. So if we have an exact triangle

E \to F \to G \to E[1]

in {\bf D}^b(Y \times Z), then we get, for any element A \in {\bf D}^b(Y), an exact triangle

\Phi^E_{Y \to Z}(A) \to \Phi^F_{Y \to Z}(A) \to \Phi^G_{Y \to Z}(A) \to \Phi^E_{Y \to Z}(A)[1]

in {\bf D}^b(Z). The key point is that in the case that Y=Z, the Fourier–Mukai transform using the structure sheaf of the diagonal \Phi_{Y \to Y}^{\mathcal{O}_\Delta} is the identity functor. To see this, let i \colon X \to X \times X be the diagonal embedding. Letting p_1,p_2 be the two projections, then

\Phi_{Y \to Y}^{{\cal O}_\Delta}(A) = {\bf R}p_{2,*}({\bf L}p_1^*A \stackrel{\bf L}{\otimes}_{{\cal O}_{Y \times Y}} i_*{\cal O}_Y)

= {\bf R}p_{2,*}({\bf R}i_*({\bf L}i^*{\bf L}p_1^*A \stackrel{\bf L}{\otimes}_{{\cal O}_Y} {\cal O}_Y) = A,

where in the last equality, we use that p_1 \circ i = p_2 \circ i is the identity map. In light of this remark and the exactness remark, it makes sense to try to find a resolution for the diagonal if we’re trying to find a set of generators of {\bf D}^b(X). So we’ll do just that.

3. Resolution of the diagonal

Let \mathcal{R} be the rank k tautological subbundle of X, and let \mathcal{Q} be the tautological quotient bundle, so that we have a short exact sequence

0 \to \mathcal{R} \to X \times V \to \mathcal{Q} \to 0.

We’ll use \boxtimes to denote the exterior tensor product of sheaves on X, i.e., if p_i \colon X \times X \to X are the two projections, and F and G are two sheaves on X, then F \boxtimes G = p_1^*F \otimes p_2^*G is a sheaf on X \times X.There is a natural map p_1^* \mathcal{R} \to V \times X \times X \to p_2^* \mathcal{Q} where the first map identifies the trivial bundle V with p_1^*V and is the corresponding inclusion, while the second map identifies V with p_2^* V and is the corresponding projection. Set-theoretically, the zero set of this map is the diagonal \Delta, and we can check locally that it also defines the diagonal scheme-theoretically.

So let s be the section of {\cal H}{\rm om}(p_1^* \mathcal{R}, p_2^*\mathcal{Q}) = \mathcal{R}^* \boxtimes \mathcal{Q} corresponding to this map. We get a Koszul complex

\displaystyle 0 \to \bigwedge^{k(n-k)} (\mathcal{R} \boxtimes \mathcal{Q}^*) \to \cdots \to \bigwedge^2 (\mathcal{R} \boxtimes \mathcal{Q}^*) \to \mathcal{R} \boxtimes \mathcal{Q}^* \to \mathcal{O}_\Delta (*)

which is exact: \Delta and X \times X are nonsingular, so \Delta is locally a complete intersection in X \times X. This idea is due to Beilinson, who did it for the case of projective space. The argument was extended to Grassmannians by Kapranov.

4. Finishing things up

Now we can splice (*) into exact triangles. To use the Fourier–Mukai transform, we have to decide if we’re going from the first factor to the second, or vice versa (there is an asymmetry in the terms of (*), so this matters). Both choices will give a different set of generators. For now, let’s go from the second factor to the first. Since the characteristic of K is zero, we get the decomposition

\displaystyle \bigwedge^i(\mathcal{R}^* \boxtimes \mathcal{Q}) = \bigoplus_{|\lambda| = i} {\bf S}_\lambda(\mathcal{R}^*) \boxtimes {\bf S}_{\lambda'}(\mathcal{Q})

where the sum is over partitions of size i, {\bf S}_\lambda denotes a Schur functor, and \lambda' is the transpose partition to \lambda. This decomposition is the dual Cauchy identity, and is a consequence, for example, of the dual Robinson–Schensted–Knuth correspondence. If the field has positive characteristic, then we only get a filtration of the LHS whose associated graded is the RHS. Of course, if k=1, this is irrelevant.

Now pick any object A \in {\bf D}^b(X). If we want to generate A, we can apply the Fourier–Mukai transform to A using the exact triangles we got by splicing (*). This shows that the set \{ \Phi^{E_\lambda}_{X \to X}(A) \} generates A, where E_\lambda = {\bf S}_\lambda(\mathcal{R}^*) \boxtimes {\bf S}_{\lambda'}(\mathcal{Q}) and the set is over all \lambda which fit in the k \times (n-k) rectangle. Ranging over all A, this is an infinite set, but we can simplify further:

\displaystyle \Phi^{E_\lambda}_{X \to X}(A) = {\bf R}p_{1,*}({\bf L}p_2^*(A) \stackrel{\bf L}{\otimes} {\bf L}p_1^*({\bf S}_\lambda(\mathcal{R}^*)) \stackrel{\bf L}{\otimes} {\bf L}p_2^*({\bf S}_{\lambda'}(\mathcal{Q})))

by definition, and using the projection formula, this is isomorphic to

{\bf S}_{\lambda}(\mathcal{R}^*) \stackrel{\bf L}{\otimes} {\bf R}p_{1,*} {\bf L}p_2^*(A \stackrel{\bf L}{\otimes} {\bf S}_{\lambda'}(\mathcal{Q})).

Using flat base change (with the notation in the above diagram, we have Z = {\rm Spec}(K), Y = X, v = p_2, and g = p_1), the above can be replaced by derived global sections:

{\bf S}_\lambda(\mathcal{R}^*) \otimes {\bf R}\Gamma(X; A \stackrel{\bf L}{\otimes} {\bf S}_{\lambda'}(\mathcal{Q})),

where the second factor is isomorphic to a cochain complex consisting of its cohomology groups with zero differentials. Hence we see that the Fourier–Mukai transform is a complex consisting of copies of {\bf S}_\lambda(\mathcal{R}^*) in various degrees. So we see that \{ {\bf S}_\lambda(\mathcal{R}^*) \} is a generating set, where \lambda ranges over all partitions fitting inside of the k \times (n-k) box.

On the other hand, if we did the Fourier–Mukai transform going from the first factor to the second factor, we would get \{ {\bf S}_\lambda(\mathcal{Q}) \} as a set of generators, where \lambda ranges over all partitions which fit inside of the (n-k) \times k box.

The last thing to check is that we get an exceptional sequence. We have a partial ordering: {\bf S}_\lambda(\mathcal{R}^*) \le {\bf S}_\mu(\mathcal{R}^*) whenever \lambda_i \le \mu_i for all i. Extending this to a total order will do the trick. To check that all of the appropriate Ext groups vanish, we need to use the Borel–Weil–Bott theorem, but I will omit this task.

References

-Steven

]]> http://concretenonsense.wordpress.com/2009/10/26/exceptional-sequences-for-the-grassmannian/feed/ 0 masnevets GLFq III: characteristic map http://concretenonsense.wordpress.com/2009/10/12/glfq-iii-characteristic-map/ http://concretenonsense.wordpress.com/2009/10/12/glfq-iii-characteristic-map/#comments Mon, 12 Oct 2009 15:47:33 +0000 Steven Sam http://concretenonsense.wordpress.com/?p=710 ]]>

In the last post of this series, I gave some definitions and facts regarding the Hall–Littlewood functions. I also sketched the relationship between symmetric functions and representations of the symmetric group. Now we’ll see how this works for the finite general linear groups.

We want to imitate the Frobenius character that is used to relate the characters of the symmetric group to the ring of symmetric functions. But since the description of the conjugacy classes of the finite general linear group (and hence the parametrization of its irreducible characters) are more complicated than the description for the symmetric group, we’ll need a bigger ring to work with.

We continue the notation from the first post. For each irreducible polynomial f \in \Phi and each positive integer i>0, we have a variable X_{i,f}, whose degree we set to be deg(f). For any symmetric function u, we set u(X_f) = u(X_{1,f}, X_{2,f}, \dots). The graded ring we work in is B = {\bf C}[e_n(X_f) \mid n \ge 1,\ f \in \Phi], where e_n denotes the elementary symmetric function. In other words, elements of B are functions which are symmetric in each family of variables X_f.

Recall from last time that for a partition \lambda, P_\lambda(x;t) and Q_\lambda(x;t) are the Hall–Littlewood and augmented Hall–Littlewood functions. We use these to define elements in B:

\tilde{P}_\lambda(X_f) = q^{-\deg(f) n(\lambda)} P_\lambda(X_f; q^{-\deg(f)})

\tilde{Q}_\lambda(X_f) = q^{\deg(f)(|\lambda| + n(\lambda))} Q_\lambda(X_f; q^{-\deg(f)}),

where n(\lambda) = \sum_i (i-1)\lambda_i. For a partition-valued function \boldsymbol{\mu}, we set

\displaystyle \tilde{P}_{\boldsymbol{\mu}} = \prod_{f \in \Phi} \tilde{P}_{\boldsymbol{\mu}(f)}(X_f),

\displaystyle \tilde{Q}_{\boldsymbol{\mu}} = \prod_{f \in \Phi} \tilde{Q}_{\boldsymbol{\mu}(f)}(X_f).

We use these two bases to define a (complex) inner product on B:

\langle \tilde{P}_{\boldsymbol{\lambda}}, \tilde{Q}_{\boldsymbol{\mu}} \rangle = \delta_{\boldsymbol{\lambda}, \boldsymbol{\mu}}.

Now we need to construct the representation ring of the finite general linear groups G_n over {\bf F}_q, where now q will remain fixed. This will be very similar to what happens for the symmetric groups. Given characters u and v for G_n and G_m, respectively, let P be the parabolic subgroup of G_{n+m} consisting of matrices of the form

g(A,B,C) = \begin{bmatrix} A & B \\ 0 & C \end{bmatrix}

where A is an n \times n matrix, B is an n \times m matrix, and C is an m \times m matrix. We define a character w on P by setting

w(g(A,B,C)) = u(A) v(B).

Then the induction product u \circ v is defined as the induced character {\rm Ind}_P^{G_{n+m}}(w). (Recall that for symmetric groups, we define the induction product by inducing from parabolic subgroups \mathfrak{S}_n \times \mathfrak{S}_m \subset \mathfrak{S}_{n+m}.) If we let A_n denote the complex vector space of characters of G_n, then the induction product gives a graded ring structure on A = \bigoplus_{n \ge 0} A_n. We can also put a complex inner product on A by setting the different graded components to be mutually orthogonal and using the standard inner product for characters on each component, just as in the case of the symmetric group. Now comes the important part: let \pi_{\boldsymbol{\mu}} denote the function which is 1 on the conjugacy class corresponding to \boldsymbol{\mu}, and 0 elsewhere. Then we have a characteristic map

{\rm ch} \colon A \to B, \quad {\rm ch}(\pi_{\boldsymbol{\mu}}) = \tilde{P}_{\boldsymbol{\mu}}.

Theorem. The characteristic map ch is an isometric isomorphism of graded rings.

If we continue with the analogy of the relationship between the symmetric group and symmetric functions, then the characteristic of the irreducible characters of G_n should be some kind of “Schur functions.” Unfortunately their definition will require significantly more notation. So I’ll skip that and just say that we can define functions S_{\boldsymbol{\lambda}}. One catch, though, is that the indexing set we use for these Schur functions is not the same as the indexing set for conjugacy classes. The indices {\boldsymbol{\lambda}} can be thought of as partition-valued functions, but on a different domain. But this is not such a big deal.

Theorem. The S_{\boldsymbol{\lambda}} form an orthonormal basis for B. Furthermore, their inverses under the characteristic map are the irreducible characters \chi^{\boldsymbol{\lambda}} of the groups G_n. Consequently, the values of the characters are given by the change of bases S_{\boldsymbol{\lambda}} = \sum_{\boldsymbol{\mu}} \chi^{\boldsymbol{\lambda}}(\boldsymbol{\mu}) \tilde{P}_{\boldsymbol{\mu}}.

At any rate, I think it is nice that the same kind of setup works for the finite general linear groups as does for the symmetric group, which maybe further justifies the statement that the finite general linear groups are q-analogues of the symmetric groups.

But since these symmetric functions are so horribly complicated, one doesn’t expect to have a nice combinatorial rule for changing from the S basis to the \tilde{P} basis (such as the Murnaghan–Nakayama rule for writing the Schur polynomials in terms of power sum symmetric functions in the symmetric group case). There are some nice cases though. When the conjugacy class \boldsymbol{\mu} corresponds to a unipotent conjugacy class, we can evaluate induced characters from maximal tori T of G_n at \boldsymbol{\mu} to get Green polynomials (up to a sign). And sometimes these induced characters are irreducible (precisely when the stabilizer of the character in the Weyl group of T is trivial).

Green polynomials are more manageable to think about: they arise as the change of basis coefficients when writing power sum symmetric functions as Hall–Littlewood functions (now working just in the ring \Lambda[t] from last time).

That’s basically all I want to say about the connection between symmetric functions and finite general linear groups. There is a more powerful approach to characters of these groups using \ell-adic cohomology due to Deligne and Lusztig, and it works more generally for any finite group of Lie type. Using that approach, it can be shown, for example, that the characters are integer valued.

-Steven

]]> http://concretenonsense.wordpress.com/2009/10/12/glfq-iii-characteristic-map/feed/ 0 masnevets A Free Association On Basic Adjoints http://concretenonsense.wordpress.com/2009/10/02/a-free-association-on-basic-adjoints/ http://concretenonsense.wordpress.com/2009/10/02/a-free-association-on-basic-adjoints/#comments Fri, 02 Oct 2009 23:52:10 +0000 yanzhang http://concretenonsense.wordpress.com/?p=713 ]]>

I’ve been betraying the title of this blog and doing some abstract nonsense lately, mostly to relearn algebraic geometry for the n-th time for some embarrassingly large n. With the memory capabilities of a muffin, I am practically starting over each time. Luckily, each adventure feels slightly more beatable (exp points?), since I have a few more examples I can use for myself. Masnevets and I had a good discussion about a few basic examples of adjoint functors (recall the definition here: basically, we need a pair of functors F \colon C \to D and G \colon D \to C such that we have a natural isomorphism \hom_D(X, F(Y)) \cong \hom_C(G(X), Y)), and thus we have a new Concrete Nonsense post.

Before we start, I want to state that I’m trying something new. This post is not intended to be an introduction to adjoints as I originally envisioned – I realized that there are many better sources for that. Instead, I’ll try to do a free association that juxtoposes a few elementary concepts. You don’t even have to know the definitions of adjoints to start seeing what I’m getting to, since I’ll be namedropping algebraic structures like Kanye West.

My first introduction to adjoints was from algebraic topology, where you naturally bump into the functors \otimes_R and \hom(R, -). It was unnecessary at the time (for the scope of the course, at least) to know that they were adjoints, but now I know them as the “tensor-hom adjunction pair” (saying “tensor-hom” a lot helps me remember tensor as the left adjoint and hom as the right). Furthermore,  knowing this relationship allows me to remember some other things – in particular, knowing the left- and right- exactness of these functors, which I used to always mix up. Left adjoints are always right-exact, and right adjoints are always left-exact. Combined with knowing that tensoring is a left-adjoint, I now know that tensoring is right-exact and adjoints are left-exact.

I’ll now make a digression pertaining to something which I believe is important but seldom discussed. By now, some readers are probably complaining that my previous statement doesn’t make it easier to remember anything, because I have to remember an equally arbitrary fact – and what if our brains find it more intuitive that “left adjoints would be left-exact,” which is wrong? In my experiences, the sets of things that are easy to remember for mathematicians are extremely different from one person to the next. Thus, it may help (especially for “elementary” topics such as this one) to just chain lots of little ideas together, so if someone links concepts A, B, and C, and Alice finds it easier to remember B and C from A, she’ll benefit just as much as Bob, who likes to remember A and C from B (and Chris is sad that nobody likes C except him). As math presentation tends to be fairly “structured” – no surprise, knowing mathematicians – I wouldn’t mind seeing more “free associations” in print or the blogosphere, because this is the form of communication which reminds me most of chatting about math with other people while drinking coffee in the common room, an experience which has often taught me a lot more than going to most lectures.

(since this is a free association, before we go back to adjoints, we might as well see another way to remember the exactness assignments from Masnevets. A good “toy exact sequence” to use here is 0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0, where the second map is multiplication by 2 and the third is taking mods . Tensoring by \mathbb{Z}/2\mathbb{Z} turns the second map into the zero map, so tensoring is not left-exact and thus must be right-exact.)

Many of you may be surprised by tensor-hom as my first choice of adjoint functors – that’s only because I learned that one first (this is a lie, and we’ll come back to later). There’s a much easier example using the forgetful functor. Consider the forgetful functor G \colon \mathrm{Grp} \to \mathrm{Set}. It happens to be the right adjoint to the functor F \colon \mathrm{Set} \to \mathrm{Grp} that sends a set to a free group generated by the set. The intuition here is this: by the definition of adjunction, we want “the number of maps out of the image of F” to be “the same” as “the number of maps into the image of G.” When G forgets so much structure, we’ll get lots of maps into the set G(Y). To get lots of maps out of F(X), we want a lot of freedom on the structure, and hence the free group. Of course, there’s nothing particularly binding here about groups; we can do this in general to get a “free functor” when we have a forgetful functor from a category of other algebraic structures to sets (we can’t do this *all* the time, but most of the time we are fine. I don’t understand the conditions here too well, which involves an overloading of the word “variety,”  so I won’t expound).

In fact, let’s take it a step further. We don’t only have to forget into sets. We can forget, for example, from abelian groups into groups, or from associative algebras into Lie algebras. What did we forget in these two cases? Commuting relations and the product, respectively. So intuitively, to “match the complexity” of the two \hom’s, we want to get just as much structure back: we want to make things commute in an arbitrary group; we want to be able to multiply things in a Lie algebra. It was quite a joyful “but of course” when I realized that the left adjoints of the two functors above came out to be abelianization and the universal enveloping algebra, respectively.

Finally, in this forgetful context I’ll return to the actual first example of adjoint functors I’ve seen (though I definitely did not know it in that context at the time). When we restrict a representation Vof G, or equivalently a F[G]-module, to a sub-representation \mathrm{Res} V of H, we’re “forgetting” how the other elements of G act on our vector space. So is there a natural way to get them back? For each representation W of of H we can induce the representation \mathrm{Ind} W. This ends up being another adjunction pair, of course. Here’s an immediate consequence: in the case that these groups are finite, note that the dimension of the two \hom’s must match; this just gives us \langle \mathrm{Ind W}, V \rangle = \langle W, \mathrm{Res} V \rangle, which is the Frobenius reciprocity formula from second-semester abstract algebra.

-Y

]]> http://concretenonsense.wordpress.com/2009/10/02/a-free-association-on-basic-adjoints/feed/ 7 KR GLFq II: Hall–Littlewood functions http://concretenonsense.wordpress.com/2009/09/28/glfq-ii-hall-littlewood-functions/ http://concretenonsense.wordpress.com/2009/09/28/glfq-ii-hall-littlewood-functions/#comments Mon, 28 Sep 2009 14:21:27 +0000 Steven Sam http://concretenonsense.wordpress.com/?p=684 ]]>

Last time, I discussed how to parametrize the conjugacy classes of the general linear group G over a finite field. In a sense, this group is a q-analogue of the symmetric group, so we might try to imitate the constructions for the symmetric group to get information about G. There’s a nice construction that Frobenius worked out which connects the characters of the symmetric group with the combinatorics of the Schur functions. I’ll briefly summarize the statement. The conjugacy classes of the symmetric group on n letters are parametrized by partitions of n. So we can also parametrize the irreducible characters by partitions as well (though it is not clear how to do this in a “canonical” way a priori). Ignoring the indexing issue (which can be dealt with) and letting \chi^\lambda(\mu) be the irreducible character indexed by \lambda evaluated at the conjugacy class consisting of permutations whose cycle lengths are given by the parts of \mu, then one has s_\lambda(x) = \sum_\mu z_\mu^{-1} \chi^\lambda(\mu) p_\mu(x) where s_\lambda(x) is a Schur function, p_\mu(x) is a power sum (Newton) symmetric function, and z_\mu = 1^{m_1} m_1! 2^{m_2} m_2! \cdots where m_i is the number of times that i appears as a part of \mu (the meaning of z_\mu is that n! z_\mu^{-1} is the size of the conjugacy class index by \mu.)

So the question to ask might be “can we find a similar interpretation for the characters of G?” The answer is yes, but becomes a bit more involved.

Instead of Schur functions, one needs to look at another class of symmetric functions called the Hall–Littlewood functions, but we’ll actually need a much larger ring than the ring of symmetric functions. While the Schur functions are symmetric functions in a set of variables x_1, x_2, \dots usually defined over rational coefficients, the Hall–Littlewood (HL) functions are symmetric functions defined over the ring {\bf Z}[t] (so t is an additional variable which does not affect the definition of “symmetric”.) Let \Lambda[t] denote the symmetric functions in variables x_1, x_2, \dots with coefficients in {\bf Z}[t]. Like the Schur functions, the HL functions are indexed by partitions. The definition of the Hall–Littlewood function indexed by \lambda in n variables x_1, \dots, x_n is

\displaystyle P_\lambda(x_1, \dots, x_n; t) = \sum_{w \in S_n / S_n^\lambda} w\left( x_1^{\lambda_1} \cdots x_n^{\lambda_n} \prod_{\lambda_i > \lambda_j} \frac{x_i - tx_j}{x_i - x_j} \right) .

Here S_n is the symmetric group on n letters, and S_n^\lambda \subseteq S_n is the subgroup of permutations w such that \lambda_{w(i)} = \lambda_i for all i. From this, one can see that P_\lambda(x_1, \dots, x_n; 1) = m_\lambda(x_1, \dots, x_n) where m_\lambda denotes the monomial symmetric function which is the sum of all of the distinct terms x_{w(1)}^{\lambda_1} \cdots x_{w(n)}^{\lambda_n} as w ranges over all permutations of n.

Set [\lambda]_t! = [\lambda_1]_t! \cdots [\lambda_n]_t!. An equivalent definition for P_\lambda is

\displaystyle P_\lambda(x_1, \dots, x_n; t) = [\lambda]_t!^{-1} \sum_{w \in S_n} w \left( x_1^{\lambda_1} \cdots x_n^{\lambda_n} \prod_{i < j} \frac{x_i - tx_j}{x_i - x_j} \right),

from which one can deduce that P_\lambda(x_1, \dots, x_n; 0) = s_\lambda(x_1, \dots, x_n) from the Weyl character formula.

I just want to state some of the properties that we will need later without giving too many details. For proofs, one can consult Macdonald's book Symmetric Functions and Hall Polynomials. From the first definition, one can deduce that these functions enjoy a stability property:

P_\lambda(x_1, \dots, x_n, 0; t) = P_\lambda(x_1, \dots, x_n; t),

and hence one can define P_\lambda(x;t) in infinitely many variables x_1, x_2, \dots by taking an inverse limit. Since the P_\lambda are symmetric functions, we can write

P_\lambda(x;t) = \sum_\mu w_{\lambda, \mu} (t) s_\mu(x)

for some polynomials w_{\lambda,\mu}(t). In fact, w_{\lambda, \lambda}(t) = 1 and w_{\lambda,\mu}(t) = 0 unless \lambda \ge \mu (dominance order), so the change of basis matrix from P_\lambda to s_\lambda is upper unitriangular, which implies that the P_\lambda(x;t) form a {\bf Z}[t]-basis of \Lambda[t].

The inverse of this change of basis is very interesting. In this case, write s_\lambda(x) = \sum_\mu K_{\lambda, \mu}(t) P_\mu(x;t). The K_{\lambda, \mu}(t) are the Kostka–Foulkes polynomials. Since P_\mu(x;1) = m_\mu(x), we see that K_{\lambda, \mu}(t) = K_{\lambda, \mu} are the Kostka numbers. It is a fact that the Kostka–Foulkes polynomials are in fact polynomials, and they have nonnegative integers. I hope to write a post about these at some point.

We will also need augmentations of these functions in the next post. First set

\displaystyle b_\lambda(t) = \prod_{i \ge 1} [(1-t)(1-t^2) \cdots (1-t^{m_i(\lambda)})]

where m_i(\lambda) is the multiplicity with which i appears in \lambda. Then define

Q_\lambda(x; t) = b_\lambda(t) P_\lambda(x; t).

Although we won’t use them, let me mention skew Hall–Littlewood functions. Define an inner product on \Lambda[t] by declaring that \langle P_\lambda(x;t), Q_\mu(x;t) \rangle = \delta_{\lambda, \mu}. Then we can define skew Hall–Littlewood functions for partitions \mu \subseteq \lambda via

\langle Q_{\lambda/\mu}(x;t), P_\nu(x;t) \rangle = \langle Q_\lambda(x;t), P_\mu(x;t) P_\nu(x;t) \rangle

\langle P_{\lambda/\mu}(x;t), Q_\nu(x;t) \rangle = \langle P_\lambda(x;t), Q_\mu(x;t) Q_\nu(x;t) \rangle.

From this definition, setting t=0 gives back the skew Schur functions s_{\lambda/\mu}(x) (since they are defined in a similar way). The weird thing, however, is that the skew Schur functions only depend on the shape \lambda/\mu, whereas the skew Hall–Littlewood functions remember both \lambda and \mu. One can write down a rather explicit formula for Q_{\lambda/\mu} in terms of semistandard tableaux which shows that the function depends on both \lambda and \mu (but this is only seen in the powers of t, and not the x_i), but I will omit this so that I can wrap this post up.

Let me just end with some other specializations of t that are important. When \lambda is a strict partition (i.e., the nonzero parts are distinct) then setting t=-1 gives P_\lambda(x;-1) = 2^{\ell(\lambda)} Q_\lambda where \ell(\lambda) is the number of parts, and Q_\lambda are the Schur Q-functions, which are important for the projective representation theory of the symmetric group (maybe a future topic). Also, specializations at t=q^{-1} for q a prime power are related to Hall algebras, which are used to keep track of extensions between finite Abelian groups.

In the next post, I’ll discuss the connection between characters of {\bf GL}_n({\bf F}_q) and symmetric functions.

-Steven

]]> http://concretenonsense.wordpress.com/2009/09/28/glfq-ii-hall-littlewood-functions/feed/ 0 masnevets GLFq I: Conjugacy classes of a finite general linear group http://concretenonsense.wordpress.com/2009/09/14/glfq-i-conjugacy-classes-of-a-finite-general-linear-group/ http://concretenonsense.wordpress.com/2009/09/14/glfq-i-conjugacy-classes-of-a-finite-general-linear-group/#comments Mon, 14 Sep 2009 15:26:32 +0000 Steven Sam http://concretenonsense.wordpress.com/?p=665 ]]>

I’d like to start a series of posts, GLFq (general linear group over a finite field), in which I hope to explain the representation theory of G = {\bf GL}_n({\bf F}_q) (n and q will remain fixed). In this first post, I’ll explain how to write down the conjugacy classes of G. In later posts, I plan to introduce Hall–Littlewood polynomials and the characteristic map. I would like to also go into how to construct the actual representations, and discuss things related to Hall–Littlewood polynomials, like the q-Kostka polynomials and a lot of the interesting algebra/geometry behind them.

There are two pieces of data we would like to know. First, what is the size of G? Second, how do we parameterize the conjugacy classes? The first question is easy to answer since an invertible matrix is given by the data of n linearly independent vectors. The first one can be chosen to be any nonzero vector, so there are q^n - 1 of them. In general, the ith one can be chosen to be any vector not in the span of the last i-1 (so we are just avoiding some i-1 dimensional subspace, which has q^{i-1} elements), and hence there are q^n - q^{i-1} choices for such a vector. All together, the number of elements of G is \prod_{i=1}^n (q^n-q^{i-1}). We can rewrite this as q^{\binom{n}{2}} (q-1)^n [n]_q! to make it more analogous to the number of elements of the symmetric group.

The second question requires the rational canonical form. If we were dealing with an algebraically closed field, conjugacy classes would of course be parameterized by Jordan normal forms, so we need some kind of substitute for that. First, we need the structure theorem for finitely generated modules over a principal ideal domain R. This says that any such module is a direct sum of its torsion submodule and a free submodule. Furthermore, the torsion submodule is uniquely a direct sum of cyclic modules, which can be written in the form R/(f^m) for some irreducible element f and some positive integer m.

Given a matrix A, we’ll apply this to the case R = {\bf F}_q[t] and the module V = {\bf F}_q^n where the action of a polynomial p(t) on V is given by p(A). Irreducible elements of R are the same as irreducible polynomials, but we will never see the polynomial x show up if A is invertible, and we will only need to use the monic ones. Let \Phi be the set of all monic irreducible polynomials over {\bf F}_q which are different from the constant polynomial x. Hence, we see that the data of the decomposition is given by a partition valued function \boldsymbol{\mu} on \Phi. Explicitly, the function \boldsymbol{\mu} gives the module \bigoplus_{f \in \Phi} \bigoplus_{i \ge 0} R / (f^{\boldsymbol{\mu}(f)_i}).

Since the dimension of R / (f^m) is \deg(f) \cdot m, the conjugacy classes of G are given by those partition valued functions \boldsymbol{\mu} such that \| \boldsymbol{\mu} \| := \sum_{f \in \Phi} \sum_i \deg(f) \boldsymbol{\mu}(f)_i = n.

As an aside, if we look at the conjugacy classes of all matrices instead of invertible matrices, i.e., we allow the domain of the partition valued functions above to include the polynomial x, then the number of such classes is \sum_j p_j(n) q^j where p_j(n) is the number of partitions of n into j parts. I think it’s a really nice formula (though it takes some work to show). See Chapter 1, Section 10 of the second edition of Enumerative Combinatorics I for a derivation of this formula.

Let’s look at the case of n=2. The only valid partition valued functions can only have nonempty values on polynomials of degree at most 2. There are 3 types of functions:

  • There exists a single monic irreducible polynomial f of degree 1 such that \mu = \boldsymbol{\mu}(f) \ne \emptyset and we have \sum_i \mu_i = 2, so \mu \in \{(2), (1,1)\}. These correspond to matrices with a single eigenvalue, the partition (2) means that it’s a diagonal matrix, and the partition (1,1) means that it is conjugate to a size 2 Jordan block.
  • There exists two distinct monic irreducible polynomials f and g of degree 1 such that \boldsymbol{\mu} takes the value (1) on both and is empty on all other polynomials. These correspond to matrices with two distinct eigenvalues.
  • There exists a single monic irreducible polynomial f of degree 2 such that \boldsymbol{\mu}(f) = (1) and all other values are the empty partition. These are matrices without a Jordan normal form.

The only irreducible polynomials of degree 1 which are allowed are of the form x-a for a nonzero value of a. So there are 2(q-1) functions of the first kind and \binom{q-1}{2} = (q-1)(q-2)/2 functions of the second kind. For the third kind, we have q^2 monic polynomials of degree 2 over {\bf F}_q. There are q polynomials of the form (x-a)^2 and \binom{q}{2} = q(q-1)/2 of the form (x-a)(x-b) for a and b distinct, so we must have q^2 - q - q(q-1)/2 = q(q-1)/2 monic irreducible degree 2 polynomials over {\bf F}_q. Thus in total we have 2(q-1) + (q-1)(q-2)/2 + q(q-1)/2 = q^2 - 1 conjugacy classes.

Next time, I’ll say something about Hall–Littlewood polynomials. Click here for the next post.

-Steven

]]> http://concretenonsense.wordpress.com/2009/09/14/glfq-i-conjugacy-classes-of-a-finite-general-linear-group/feed/ 0 masnevets Symplectic geometry II http://concretenonsense.wordpress.com/2009/09/03/symplectic-geometry-ii/ http://concretenonsense.wordpress.com/2009/09/03/symplectic-geometry-ii/#comments Thu, 03 Sep 2009 04:31:01 +0000 lewallen http://concretenonsense.wordpress.com/?p=661 ]]>

Back to symplectic geometry. So far, everything I did in my last post only used the fact that the symplectic form {\omega} was skew symmetric, not that it was closed. Indeed the “closed” property is rather mysterious, (as far as I’m concerned, although in the literature it is called “geometric”), since I don’t know of any really good geometric intuition for the action of exterior derivative on 2-forms. Still, it is a hugely important condition, and key to many of the special properties of symplectic geometry, notably for us, the Darboux theorem. Note that there is no real equivalent condition for Riemannian structures, and therefore it takes us in a whole new direction. I would love to have a better sense of how to “explain” why certain symplectic arguments don’t work in the Riemannian world (eg Darboux theorem), but I haven’t delved deeply enough into the proof to do this, since it’s not coherent to just say “the Riemannian form isn’t closed.”

For an application of closedness, recall the Lie derivative {L_{X}v} of a tensor {v} with respect to a vector field {X}: it’s just the derivative of {v} along the flow lines of {X}. In the case that {v} is a differential form, one can prove formally, using induction and certain algebraic properties of {L_{X}v}, that

\displaystyle L_{X}v =d i_{X}(v) + i_{X}dv

where {i_{X}} of an {n}-form is just contraction of {X} with that form (ie evaluate that form on {X} to get an {n-1} form). Again, this formula means nothing to me other than some algebra (I wish I could change that!), but it’s called “Cartan’s (magic) formula” and it’s very nice. In particular, suppose {X= X_{f}} is Hamiltonian, and we want to evaluate {L_{X_{f}}\omega}; contraction of {X_{f}} and {\omega} gives {df} by definition which is exact and therefore closed; what’s more, because {\omega} is closed, the other term in Cartan’s formula is 0, and we see that {L_{X}\omega=0}—the symplectic form is invariant under all Hamiltonian flows, which is to say, the flow {\rho_{t}} of {X_{f}} is a one parameter group of symplectomorphisms (diffeomorphisms which preserve the symplectic form, the analogy of isometries in Riemannian geometry)! The equivalent concept for Riemannian geometry is a Killing vector field, and generically, these are very hard to find, which one can see because a generic Riemannian manifold has a finite group of isometries (right?). But in the symplectic case we’ve just produced lots of (one-parameter subgroups of!) symplectomorphisms, indeed the space is infinite dimensional, and in this way is much more akin to volume preserving smooth maps (of which it is a subset, since {\omega^{n}} is a volume form). In particular, we’ve shown that for every smooth function {f} we get a family of symplectomorphisms, tangent to X_{f}. In general a vector field whose flow is a family of symplecticmorphisms is called symplectic—by Cartan’s formula, this is the case if contracted with {\omega}, it yields a closed 1-form—and we see that Hamiltonian vector fields correspond exactly to the case that this closed form is exact.

Questions to do with Hamiltonian symplectomorphisms (those that come from Hamiltonian vector fields), and related subjects, are at the heart of symplectic geometry. For example, one can ask about their fixed points, which is the Arnold conjecture, and there is the related Weinstein conjecture, just proved by Taubes. For now I won’t go into these, and will let the comparison with Riemannian geometry be the main motivation.

Concerning that comparison, we remarked earlier that on a fixed vector space {{\mathbb R}^{2n}}, both symmetric and skew-symmetric forms, when non-degenerate, can always be put into a standard form, in other words, their only invariant is the dimension of the underlying space. Therefore at any point in our manifold, we know what the form looks like at that point. However, in Riemannian geometry, if we could construct some neighborhood around a point for which, in local coordinates, the metric restricted to every point in that neighborhood was standard, then we would have shown that that neighborhood was isometric to Euclidean space, and therefore flat, for example. But not all Riemannian manifolds are locally flat! Thus even locally, Riemannian geometry is quite complex. The same thing, remarkably, is NOT true for symplectic manifolds. This is Darboux’s theorem.

The proof I will give starts with a lemma called Moser’s trick; there is also a more hands on and dirty, “down-to-earth geometric” proof, I believe. Anyway, suppose {M} is compact, and we have a smooth family of symplectic (closed, non-degenerate) forms {\omega_{t}\in \Omega^{2}(M)}, {t\in [0,1]} (here {\Omega^{2}(M)} is the space of differential 2-forms), with an “exact derivative,” meaning that if we look at {d/dt (w_{t})} for each {t} (which we can define in the usual way, since {\Omega^{2}(M)} is an {{\mathbb R}}-vector space), then the resulting 2-form can be written as {d\sigma_{t}} for some smooth family {\sigma_{t}} of 1-forms. This is the assumption. Then Moser tells us that this path of 2-forms, which lives only in some associated space of forms, can actually be realized geometrically on {M} via a family of diffeomorphisms ({\phi_{t}} such that {\phi_{t}^{*}(\omega_{t})=\omega_{0}}) (NOT symplectomorphisms, duh, since they change the symplectic form). The proof is quick: each {\omega_{t}} is non degenerate, so it gives us a vector field {X_{t}} for each {t}, which is just the contraction {\tilde{\omega_{t}}(\sigma_{t})} (as an anecdote, note here that {\omega_{t}} is actually {d\sigma_{t}}, so we’re plugging the primitive into its own derivative! This always seemed a little incestual to me. On a more serious note, it comes up all the time: is there some more concise or intuitive way to describe what it’s measuring?). Since {M} is compact, we have a flow {\rho_{t}} of diffeomorphisms generated by {X_{t}}. And this is the desired family of maps! Because

\displaystyle d/dt(\rho_{t}^{*}(w_{t}))=\rho_{t}^{*}(L_{X_{t}}\omega_{t})+\rho_{t}^{*}(dw_{t}/dt)=\rho_{t}^{*}(d i_{X_{t}}\omega_{t}+d\omega_{t}/dt) =0

for all {t}! (the last equality follows from the linearity of {\rho^{*}}, and the first is one of these fancy Lie-derivative identities (applied to the flow of a time-dependant vector field) which one simply proves directly). So it’s constant, and it starts at {\omega_{0}} (because {\rho^{*}_{0}} is the identity).

Note that the assumption that {d/dt (\omega_{t})} is exact implies that {[w_{t}]\in H^{2}(M)} is constant. Indeed, this latter assumption is actually sufficient for the theorem. So this says that any two “isotopic forms” (cohomologous symplectic forms connected by a path of cohomologous symplectic forms) are actually “strongly isotopic,” meaning the path of forms can be realized by a path of diffeomorphisms, as above.

Finally we use Moser’s trick to prove Darboux’s theorem. Let {m\in M}, here {M} is not necessarily compact. Choose a basis in {T_{m}M} in which {\omega} is standard (just at {T_{m}M)}. Now we can extend the coordinates {(X_{1},\dots X_{n},Y_{1},\dots Y_{n})} in {T_{m}M} to a local coordinate patch around {m}, and we have two forms: the form {\omega_{0}=\omega}, the one we started with, and the form {\omega_{1}} defined to be the standard form at every point (note that by construction, these two forms agree at {p}). Now for our family of forms interpolating, we just take the line {\omega_{t}=t\omega_{1}+(1-t)\omega_{0}}. We want to make sure these forms are non-degenerate (they are obviously closed), but because non-degeneracy is an open condition in the space of 2-forms, we can take our coordinate chart {U} small enough so that we’re ok. Now note that the {t}-derivative of {\omega_{t}} is just {\omega_{1}-\omega_{0}} independently of {t}; this form is closed, so locally it is exact (we’re doing everything locally) so let {\alpha} be an anti-derivative. Fixing up {\alpha} by a constant, we can assume it’s 0 at {p}. Now we define {{X}_{t}} to be the vector field which is {\alpha} contracted with {{\omega}_{t}}. By restricting ourselves (again!) to a small set inside of our chart, we can define a global flow for {{X}_{t}} (so this is slightly different than Moser’s trick, which was global but for which we used compactness of {M}). So indeed, applying the reasoning from Moser’s trick, {\rho^{*}_{t}} interpolates between {w_{0}} and {w_{1}}, in particular, {\rho_{1}^{*}\omega_{1} = \omega_{0}}, which is the desired result. What’s more, because we fixed things up so that {\alpha}, and therefore its dual vector field {{X}_{t}}, was zero at {p}, the form {\omega} is actually constant at {p} as we apply the diffeomorphisms. Q.E.D.

Next time: ? Almost complex structures, contact geometry, J-holomorphic curves, examples, knot theory, something completely unrelated, nothing?

]]> http://concretenonsense.wordpress.com/2009/09/03/symplectic-geometry-ii/feed/ 6 lewallen Symplectic geometry I http://concretenonsense.wordpress.com/2009/09/02/638/ http://concretenonsense.wordpress.com/2009/09/02/638/#comments Wed, 02 Sep 2009 21:35:57 +0000 lewallen http://concretenonsense.wordpress.com/?p=638 ]]>

One of my summer projects was to try to learn symplectic geometry. In this, my first installment of notes, I discuss some introductory notions; hopefully it’s not too rambling. In the continuation, I’ll prove Darboux’s theorem, a fundamental result which says that locally, all symplectic spaces are isomorphic (something which sharply distinguishes symplectic geometry from Riemannian geometry, where there are many local invariants, such as curvature).

EDIT: Here, I take the point of view that the reader is somewhat familiar with Riemannian geometry, and try to build intuition for the structures in symplectic geometry via an analogy with the Riemannian case. This was helpful for me, to some extent, in order to even have a chance of “breaking into” the field, so to speak. However, there are many reasons why this is possibly a misleading vantage point, so do not believe that it’s the whole story. It may be helpful for some. Please see the comments for additional (undoubtedly better, I am an extreme novice) points of view. Some of these would also be quite suitable for an introduction.

In both symplectic and Riemannian geometry, the main object of study is a smooth manifold equipped with a bilinear form on each tangent space, in such a way that the forms vary smoothly as we move between tangent spaces. In the (possibly more familiar) Riemannian case, this form is a symmetric, non-degenerate, positive definite form, turning each tangent space into a normed vector space. In symplectic geometry, we instead require a skew-symmetric bilinear form on each tangent space, again varying smoothly. We still require that at each point {m} in our manifold {M}, {\omega_{m}} should be non-degenerate, so that if {\omega_{m}(X,Y)}=0 for all {Y\in T_{m}M}, then {X} must be 0. Finally, note that because {\omega} is a skew-symmetric 2-form, it is a differential 2-form on {M}, and we require that as a 2-form, {\omega} is closed, i.e., {d\omega = 0}. I’ll introduce examples as we go.

Just like in Riemannian geometry, the fact that the symplectic form is non-degenerate establishes a natural bijection between the tangent space and cotangent space at every point. I’ll write this as {\tilde{\omega}: T_{m}M\rightarrow T_{m}^{*}M}. The image of {X} is the functional which sends {Y} to {\omega(X,Y)}. In the Riemannian geometry case, we know that, at a given point, we can just choose an orthonormal basis, and then we have a particularly nice “dual basis:” a basis vector {X_{i}} is dual to the unique functional whose value on that vector is 1, and whose value on the other basis vectors is 0. This is a simple way to get a handle on the isomorphism from {T_{m}M\rightarrow T_{m}^{*}M} in the Riemannian case. In terms of this special basis, the symmetric form has been reduced to the standard dot product.

There is an equivalent construction in the symplectic case. The “standard” symplectic form on {{\mathbb R}^{2n}} has a basis of the form {\left\{X_{1},\dots, X_{n}, Y_{1},\dots , Y_{n}\right\}}, with {X_{i}} pairing with {Y_{i}} to give 1 and pairing with the rest of the basis vectors (including itself) to give 0 (the form is determined by this data, along with the requirement that it be bilinear and skew-symmetric). There is a theorem, just as easy as Gram-Schmidt, that says that every symplectic form looks like the above in an appropriate basis (a corollary is that if a vector space can be equipped with a non-degenerate skew-symmetric form, then it is necessarily even-dimensional). Therefore after a change of basis, the bijection {\tilde{\omega}} between tangent and cotangent space sends {X_{i}} to the covector (linear functional) which is 1 on {Y_{i}} and 0 on the other basis vectors. As a side note, if we make {{\mathbb R}^{2n}} into a complex vector space by letting {Z_{i}=X_{i}+iY_{i}}, then the standard form {\omega} can be written in terms of the standard dot product as {\omega(A,B)=\langle iA,B\rangle}, indeed, we have {iX_{i}=Y_{i}} (I hope I have my signs ok). Thus the standard complex, Riemannian, and symplectic structures on {{\mathbb R}^{2n}} all determine each other in this way, and make it into a Kähler manifold. More generally, complex or almost-complex structures compatible with symplectic forms are very important in the subject.

Before I move on to symplectic manifolds, note that just on the level of bilinear forms on vector spaces, there are many features of the geometry of skew-symmetric forms which are quite different than that of symmetric forms (to which we are probably more accustomed), and these all have important implications in the non-linear (= general symplectic geometry) case. For example, for {S\subset {\mathbb R}^{2n}}, define {S^{\perp}=\{ X: \omega(X,Y)=0 \text{ for all } Y\in S\}}. Then the non-degeneracy of the form insures that, just like for a normed vector space, the dimensions of {S} and {S^{\perp}} add up to {2n}. However they are not necessarily disjoint! Indeed, for example, with the standard form, {\text{Span}(X_{1},\dots, X_{n})^{\perp}=\text{Span}(X_{1},\dots, X_{n})}. Such a subspace is called Lagrangian, meaning the form restricted to it is identically 0, and it is of maximal dimension ({=n}) with this property.

I want to give some examples before I go further, but unfortunately, I haven’t really found any amazingly clear and enlightening ones. The simplest and most obvious is {{\mathbb R}^{2n}} itself, with the standard form, which we have already discussed. Already in this case there are interesting questions to be asked. As another, note that if {S} is a smooth orientable surface, then every 2-form is closed, as there are no non-zero 3-forms, and any volume form will be non-degenerate. Therefore all surfaces are symplectic manifolds, and any one-dimensional subspace will be Lagrangian. So until we either introduce some more interesting questions, or go to higher dimensions, we are a little stuck (I could also mention here that every cotangent bundle is a symplectic manifold in a natural way, but I’ll probably talk about this next time). The second thing I want to mention, because it bothered me at first, is the question of motivation for symplectic geometry. Certainly, the formalism arose from physics, and simplified many natural physical models. However, I like to think of it as just another “kind” of geometry on a smooth manifold, another fairly simple structure we can put on it—an evil twin of Riemannian geometry, for which we can ask all the same questions, and see what comes out. This point of view then justifies itself as we find quite elegant behavior, as well as applications to other fields (which I probably won’t discuss this time around).

I have to make a quick disclaimer right now. Throughout, whenever I talk about integrating a vector field to a flow, I’m going to assume implicitly (without stating!) that we’re on a compact manifold {M}, so that the flow is well defined on all of {M}. This is very sloppy: might as well call the rest of these notes “Stuff about compact symplectic manifolds.” Except that Darboux’s theorem is true in general, which I’ll indicate.

OK, on to geometry. To every smooth real-valued function {f} on a smooth manifold {M}, we can associated the differential {df}, which is the one-form whose value at a vector field {X} at a point {m\in M} is the value of the directional derivative {X(f)} at {m}. The nice thing about having the dual map {\tilde{\omega}} at each point is that from each differential 1-form we can produce a vector field, and this applies to {df}, giving a vector field {\tilde{\omega}^{-1}(df):=X_{f}} which depends on {f}. Now when we do this construction with a Riemannian metric, we get (by definition) the gradient vector field, and it has the nice property that it is normal to the level sets of {f} (indeed, it points towards the direction of most increase of {f}, which naturally is perpendicular to the level sets). To see this, one just applies the derivation {\text{Grad}(f)} to {f}, and unwinding the definitions one gets simply the smooth function {\langle \text{Grad}(f)_{m},\text{Grad}(f)_{m}\rangle }, the norm squared of {\text{Grad}(f)_{m}} for each point {m\in M}. However, doing this with {X_{f}} gives the directional derivative {X_{f}(f)} as {\omega(X_{f},X_{f})=0}! Indeed, {X_{f}} is tangent to the level sets of {f} (so it is just {\text{Grad}(f)} rotated by {\pi/2} in some direction, or, alternatively, hit by the multiplication by {i} map, in an appropriate basis and some compatible complex structure). So the flow of any point {m} under {X_{f}} stays within a particular level set, and so the level sets of {f} are invariant under the flow (the standard example is to take the sphere {S^{2}} with its standard volume form as the symplectic form, and take the function {f} to be the height function. Then the level sets are circles of constant height, and the flow of {X_{f}} can be seen to be rotation around the vertical axis). This is perhaps the moment to mention that {X_{f}} is called a Hamiltonian vector field (rather than gradient vector field as in Riemannian geometry), and {f} is called the Hamiltonian (function) (of {X_{f})}. Indeed, the fact that the level sets of {f} are invariant under the flow is a hint at the physics origins (as is the name Hamiltonian) of the whole subject. Originally, one would take a particular Hamiltonian {H} to be the energy on a particular phase space (which is itself defined as the cotangent bundle to a state space, giving it a natural symplectic structure, as I alluded to previously). Then one declares that the allowable evolution of the universe is to follow the flow of {X_{H}}. That level sets are preserved under the flow gives conservation of energy. This formalism is called Hamiltonian mechanics.

To be continued! I’ll say a little more about Hamiltonian flows, and then prove Darboux’s theorem.

]]> http://concretenonsense.wordpress.com/2009/09/02/638/feed/ 6 lewallen A Fock space representation http://concretenonsense.wordpress.com/2009/08/31/a-fock-space-representation/ http://concretenonsense.wordpress.com/2009/08/31/a-fock-space-representation/#comments Mon, 31 Aug 2009 13:39:23 +0000 Steven Sam http://concretenonsense.wordpress.com/?p=623 ]]>

Lately I’ve been reading about quantum groups. I particularly enjoyed Bernard Leclerc’s paper Symmetric functions and the Fock space representation of U_q(\widehat{\mathfrak{sl}_n}), so I want to discuss a little bit of it. In particular, I want to give an example of the technique of deforming an object in order to see “hidden structure.” My notation will differ slightly from Leclerc’s since he uses the French notation for Young diagrams.

One thing that has always been really hard for me to wrap my head around is the really complicated presentations that affine Lie algebras have and how one is supposed to do anything with them. This post will be about the affine Lie algebra \widehat{\mathfrak{sl}_n}, which is defined in the paper as the Lie algebra with generators e_i, f_i, h_i (0 \le i \le n-1) and d, with 5 lines of relations. Let K be a field of characteristic 0. Another way around this is to first define the loop algebra L(\mathfrak{g}) of a simple Lie algebra \mathfrak{g} as: L(\mathfrak{g}) = \mathfrak{g} \otimes K[t,t^{-1}] with a Lie bracket given by [a \otimes t^n, b \otimes t^m] = [a,b]_{\mathfrak{g}} \otimes t^{n+m}, and then to say that \widehat{\mathfrak{g}} is its universal central extension. More precisely, we say add a central element c, and then extend the bracket above via [a \otimes t^n, b \otimes t^m] = [a,b]_{\mathfrak{g}} \otimes t^{n+m} + (a,b)_{\mathfrak{g}} n \delta_{n,-m} c where \delta is the Kronecker delta, and (,) is the Killing form of \mathfrak{g}.

In the case that \mathfrak{g} is \mathfrak{sl}_n, I want to discuss a more concrete (combinatorial) description. Just as \mathfrak{sl}_n can be thought as the traceless operators on an n-dimensional vector space (the standard representation), we can also find a standard representation for \widehat{\mathfrak{sl}_n} (the Fock space representation). For this, let {\cal F} denote the ring of symmetric functions over K in infinitely many variables. The Schur functions s_\lambda form a basis indexed by partitions, and this will be our representation. In order to describe the actions of \widehat{\mathfrak{sl}_n} on Sym, we’ll need some notation.

First, we represent partitions \lambda by their Young diagram (\lambda_i boxes drawn in the ith row, left justified). The content of a box (i,j) is the number i-j. We’ll consider contents modulo n, and say that a box is an i-node if it has content i modulo n. We define e_is_\lambda = \sum_\mu s_\mu (resp. f_is_\lambda = \sum_\mu s_\mu) where the sum is over all \mu obtained from \lambda by removing (resp. adding) an i-node, and define ds_\lambda = N_0(\lambda)s_\lambda where N_0(\lambda) is the number of 0-nodes of \lambda. Finally, set h_i = e_if_i - f_ie_i. Then \widehat{\mathfrak{sl}_n} is the Lie algebra spanned by these generators.

Unlike the \mathfrak{sl}_n case, the Fock space representation is not irreducible. Let p_\lambda = \prod_i p_{\lambda_i} be the power sum symmetric function where p_i = \sum_{j \ge 1} x_j^i. It turns out that the set \{p_{n\lambda}\} where n\lambda = (n\lambda_1, n\lambda_2, \dots) are the highest weight vectors of this representation (i.e., they are killed by the e_i, and are eigenvectors for d and the h_i). Furthermore, one has dp_{n\lambda} = |\lambda| p_{n\lambda}, so we have a natural notion of degree for our highest weight vectors. Within these graded subsets, the p_{n\lambda} give an obvious choice of basis, but there is no reason to favor them: for example, \{ p_{(2n)} + p_{(n,n)}, p_{(2n)} - p_{(n,n)} \} also forms a basis for the highest weight vectors of degree 2. The point is that the Schur functions give a “natural basis” for {\cal F} in the sense that we have defined our operators in this basis, and the basis of highest weight vectors should have “nice” properties with respect to this fixed basis, although it’s not clear what nice means right now.

The next part is something that I am learning to appreciate: since there is no way to figure out a “canonical basis” for the highest weight vectors, we should introduce a new parameter to make the structure of the Fock space representation more rigid. This new parameter is made precise by replacing \widehat{\mathfrak{sl}_n} by its q-analogue U_q(\widehat{\mathfrak{sl}_n}), and similarly for {\cal F}. More precisely, we can’t deform the Lie algebra, but we can deform its universal enveloping algebra. The quantum group U_q(\widehat{\mathfrak{sl}_n}) has generators E_i, F_i, K_i, K_i^{-1}, and D, D^{-1} and even more relations than \widehat{\mathfrak{sl}_n} has, so rather than give those, I just want to mention how to change the action on {\cal F}. Let q be a transcendental element over K, and let K(q) be the function field over K. We set {\cal F}_q = {\cal F} \otimes_K K(q), and to get the actions of the E_i and F_i on {\cal F}_q, we’ll use almost the same formulas as above, but we’ll need a bit more partition notation.

Given a partition, a box is removable (resp. indent) if it can be removed (resp. added) to obtain another Young diagram. Let \lambda and \mu be two partitions such that \lambda is obtained from \mu by adding an i-node \gamma to it. Let I^r_i(\lambda, \mu) (resp. R^r_i(\lambda, \mu)) be the number of indent i-nodes of \lambda (resp. number of removable i-nodes of \lambda) which are strictly to the right of \gamma. Also set N^r_i(\lambda, \mu) = I^r_i(\lambda, \mu) - R^r_i(\lambda, \mu). Define the same numbers with the superscript r replaced by l by replacing “right” with “left.” Then we set F_is_\lambda = \sum_\mu q^{N_i^r(\lambda, \mu)} s_\mu (resp. E_is_\nu = \sum_\mu q^{-N_i^l(\mu, \nu)} s_\mu) where the sum is over all \mu such that \mu / \lambda is an i-node (resp. \mu / \nu is an i-node). We also set D^{\pm}s_\lambda = q^{\pm N_0(\lambda)}s_\lambda and define K^{\pm}_is_\lambda = q^{\pm K(i, \lambda)}s_\lambda where K(i, \lambda) is the number of removable and indent i-nodes of \lambda. And we can take the quantum group U_q(\widehat{\mathfrak{sl}_n}) to be the K(q) algebra spanned by these generators.

We’ll use a family of operators V_k to find a nice basis. To define their actions, we need some definitions about ribbons. First, an m-ribbon is a connected skew Young diagram with m boxes which does not contain a 2 \times 2 square. The most northeast box of an m-ribbon is called its origin. Its spin is the number of rows it has minus 1. A connected union of m-ribbons is a horizontal m-ribbon strip if it is a skew Young diagram, and if the origin of each ribbon does not lie below another box in the same column. The weight of a horizontal m-ribbon strip is the number of m-ribbons used to build it. Any tiling of a horizontal m-ribbon strip subject to these constraints is unique, so we can define the spin of a horizontal m-ribbon strip to be the sum of the spins of its ribbons. We define V_k s_\lambda = \sum_\mu (-q)^{{\rm spin}(\mu/\lambda)} s_\mu where the sum is over all \mu such that \mu / \lambda is a horizontal n-ribbon strip of weight k. This sort of looks like the definitions one uses to define the Murnaghan–Nakayama rule for multiplying a Schur function and power sum symmetric function. In fact, in the classical limit “q=1″, V_k reduces to multiplication by the plethysm h_n \circ p_k = h_n(x_1^k, x_2^k, \dots ).

We introduce a K-linear bar involution on K(q) via q \mapsto q^{-1}, and then extend this to a compatible K-linear involution x \mapsto \overline{x} on {\cal F}_q by requiring that it commute with the actions of F_i and V_k on {\cal F}_q, and that it fixes the basis vector s_\emptyset. Let L (resp. L^-) be the free {\bf Z}[q]-submodule (resp. {\bf Z}[q^{-1}]-submodule) of {\cal F}_q spanned by the basis \{s_\lambda\}. Then we have the following theorem.

Theorem. There exist two unique bar-invariant bases B = \{G(\lambda)\} and B^- = \{G^-(\lambda)\} of {\cal F}_q such that G(\lambda) = s_\lambda \pmod {qL} and G^-(\lambda) = s_\lambda \pmod {q^{-1}L^-}.

The two bases are called the canonical basis and dual canonical basis of {\cal F}_q. They have a lot of nice properties. Going back to highest weight vectors, it turns out that G^-(n\lambda) is a highest weight vector for all \lambda. Furthermore, “setting q=1” this basis of highest weight vectors reduces to the plethysms s_\lambda \circ p_n = s_\lambda(x_1^n, x_2^n, \dots) (this is related to the classical limit of the operators V_k). Since it comes from a more “rigid” basis, we might be satisfied with this choice for a basis of highest weight vectors in {\cal F}. Another nice property which happens with canonical bases is a nonnegativity property: write G(\mu) = \sum_\lambda d_{\lambda, \mu}(q) s_\lambda and G^-(\lambda) = \sum_\mu e_{\lambda, \mu}(-q^{-1}) s_\mu where the d and e are polynomials.

Theorem.The polynomials d and e have nonnegative coefficients as polynomials in q. Furthermore, d_{\lambda, \mu}(q) is nonzero only if \lambda \le \mu and similarly, e_{\lambda, \mu}(q) is nonzero only if \mu \le \lambda.

Here we are using the dominance order on partitions: \lambda \le \mu if \lambda - \mu can be written as a nonnegative linear combination of vectors \varepsilon_i - \varepsilon_{i+1} where \varepsilon_i is the vector with a 1 in the ith coordinate and 0s in the other coordinates.

There is a bunch of other stuff which Leclerc discusses in the paper, like connections to Kazhdan-Lusztig polynomials and Macdonald polynomials, which illustrates why these canonical bases and their change of basis matrices are important, but I’ll stop here.

-Steven

]]> http://concretenonsense.wordpress.com/2009/08/31/a-fock-space-representation/feed/ 5 masnevets Trees, The BEST Theorem, and Alexander Polynomials http://concretenonsense.wordpress.com/2009/08/20/trees-the-best-theorem-and-alexander-polynomials/ http://concretenonsense.wordpress.com/2009/08/20/trees-the-best-theorem-and-alexander-polynomials/#comments Thu, 20 Aug 2009 03:39:20 +0000 yanzhang http://concretenonsense.wordpress.com/?p=619 ]]>

Most of my “free math time” has been used to study for quals, but today I’ve made myself post to stop Steven from taking over this blog.

One of my favorite elementary algebric combinatorial results is the Matrix Tree Theorem, which states:

In a nondirected graph with vertices labelled 1, 2, \ldots n, the number of spanning trees is equal to any principal minor of the Laplacian.

This cute result gets the number of trees on n vertices (n^{n-2}) fairly quickly with some matrix manipulation, which I will leave as an exercise to the reader. I know two proofs of this theorem: the first one involves using the Cauchy-Binet formula on the Laplacian L, after making the slick observation that L = MM^t, where M is the incidence matrix. Another quick solution can be obtained by invoking the lesser-known version of the Matrix Tree Theorem for directed graphs, which is actually a bit simpler to prove:

In a directed graph with vertices labelled 1, 2, \ldots n, the number of arborescences into vertex i (that is, trees rooted at i where all the edges point towards i) is equal to the i-th minor of the Laplacian.

But this is not all!

I actually recently learned of a third proof (involving Gessel-Viennot, of all things), but I will not mention it here (see the second reference of this post). The reason I mention the directed version and arborescences is to introduce a lesser-known but closely related result to the Matrix Tree Theorem, the forcibly-named BEST Theorem (‘B’ for de Bruijn, ‘S’ for Smith, ‘T’ for Tutte, and ‘E’ for… van Ardenne-Ehrenfest):

In a balanced directed graph (that is, for each vertex the out- and in-degrees equal) with vertices labelled 1, 2, \ldots n, the number of Eulerian circuits is equal to T \prod_v (d_v - 1)!, where d_v is the outdegree of vertex v and $T$ is the number of directed arborescences into any vertex.

This result is neat, for a couple of reasons. One, it shows immediately that the number of directed arborescences into any vertex in a balanced graph is equal, which is totally not obvious. Second, it implies a connection between Eulerian circuits and spanning trees, which is counterintuitive; in fact, when each vertex has in- and out- degree 2, which is a kind of graph we get quite often (especially when we consider nondirected graphs as directed graphs), we get that the number of Eulerian circuits is exactly the number of arborescences.

Sketch of Proof: pick any edge (i, j) to start with. Now, from any Eulerian circuit C, construct a directed graph T' as follows: for each vertex v (besides i), consider the last step in the circuit away from it towards a vertex j. Then add the directed edge (v, j) to T'. This is an arborescence into i. Note that besides the last exit from v (which must be to j), the other d_v - 1 exits can be done in any order. This creates a \prod_v (d_v - 1)! – fold bijection between arborescences into i and Eulerian circuits, which is exactly what we need.

Those familiar with knot theory may recall the Alexander polynomial \Delta_L(t) of a link L, which one obtains as a minorof a matrix M_D constructed from any diagram D of L (though it does not depend on the diagram). I’ll not review the construction since it is best done by example, it is a little tedious to type, and I’m too lazy to draw pictures (though there’s a functional Wikipedia page here). Now, it is well known that |\Delta_L(-1)| is well-defined (and called the determinant of L). However, it has a combinatorial meaning. Construct the following graph on the strands 1, 2, \ldots, n of L: each time the strand i is crossed above by j and comes out the other side as k, draw directed edges (i, j) and (i, k) (I think this even works if $i = j$, though I believe (I don’t know much knot theory and this is pure intuition, so correct me if I’m wrong!!!) you can always draw diagrams to avoid this). It takes a bit of thinking, but convince yourself that this creates a graph G with indegree and outdegree 2 at each vertex. Thus, by considering the construction of M_D, we have:

|\Delta_L(-1)| equals any minor of M_D(-1), which in turn equals both the number of arborescences into any vertex of G and the number of Eulerian circuits of G.

This is basically all I know about knot theory, so I’ll stop here.

References: Enumerative Combinatorics Vol. 2 (Stanley) for the Matrix Tree Theorem and the BEST Theorem, and A Course in Enumeration (Aigner) for the BEST Theorem and Alexander polynomials.

-Y

P.S. The latter reference deserves some mention, because it has a neat presentation. At the end of each chapter, Aigner gives a “highlight” section with a particularly pretty result (the BEST Theorem being one of them), which serves as fun enrichment material. I wish more math books did this (though this is not the first book I know which does this – Alon’s The Probabilistic Method also has similar sections, which were equally delightful). The exposition is quite good, coming from one of the writers of the beautiful Proofs from the BOOK.

]]> http://concretenonsense.wordpress.com/2009/08/20/trees-the-best-theorem-and-alexander-polynomials/feed/ 5 KR